
theorem Th74:
  for F1, F2 being non empty one-to-one Graph-yielding Function
  st F1, F2 are_Disomorphic holds rng F1, rng F2 are_Disomorphic
proof
  let F1, F2 be non empty one-to-one Graph-yielding Function;
  assume F1, F2 are_Disomorphic;
  then consider p being one-to-one Function such that
    A1: dom p = dom F1 & rng p = dom F2 and
    A2: for x being object st x in dom F1 ex G1, G2 being _Graph
      st G1 = F1.x & G2 = F2.(p.x) & G2 is G1-Disomorphic;
  reconsider f = F2*p*(F1") as one-to-one Function;
  take f;
  A3: dom(F2*p) = dom p by A1, RELAT_1:27
    .= rng(F1") by A1, FUNCT_1:33;
  hence dom f = dom(F1") by RELAT_1:27
    .= rng F1 by FUNCT_1:33;
  thus rng f = rng(F2*p) by A3, RELAT_1:28
    .= rng F2  by A1, RELAT_1:28;
  let G be _Graph;
  assume G in rng F1;
  then consider x being object such that
    A4: x in dom F1 & F1.x = G by FUNCT_1:def 3;
  consider G1, G2 being _Graph such that
    A5: G1 = F1.x & G2 = F2.(p.x) & G2 is G1-Disomorphic by A2, A4;
  F1.x in rng F1 by A4, FUNCT_1:3;
  then A6: G in dom(F1") by A4, FUNCT_1:33;
  then G in dom f by A3, RELAT_1:27;
  then A7: G in dom(F2*(p*(F1"))) by RELAT_1:36;
  G2 = F2.(p.(F1".(F1.x))) by A4, A5, FUNCT_1:34
    .= F2.((p*(F1")).G) by A4, A6, FUNCT_1:13
    .= (F2*(p*(F1"))).G by A7, FUNCT_1:12
    .= f.G by RELAT_1:36;
  hence f.G is G-Disomorphic _Graph by A4, A5;
end;
